A Lagrangian cylindrical coordinate system for characterizingdynamic surface geometry of tubular anatomic structures

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Citation for the original published paper (version of record):Lundh, T., Suh, G., DiGiacomo, P. et al (2018)A Lagrangian cylindrical coordinate system for characterizing dynamic surface geometry oftubular anatomic structuresMedical and Biological Engineering and Computing, 56(9): 1659-1668http://dx.doi.org/10.1007/s11517-018-1801-8

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Medical & Biological Engineering &Computing ISSN 0140-0118Volume 56Number 9 Med Biol Eng Comput (2018)56:1659-1668DOI 10.1007/s11517-018-1801-8

A Lagrangian cylindrical coordinatesystem for characterizing dynamic surfacegeometry of tubular anatomic structures

Torbjörn Lundh, Ga-Young Suh, PhillipDiGiacomo & Christopher Cheng

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ORIGINAL ARTICLE

A Lagrangian cylindrical coordinate system for characterizing dynamicsurface geometry of tubular anatomic structures

Torbjörn Lundh1,2& Ga-Young Suh1

& Phillip DiGiacomo3& Christopher Cheng1

Received: 26 May 2017 /Accepted: 3 February 2018 /Published online: 3 March 2018# The Author(s) 2018. This article is an open access publication

AbstractVascular morphology characterization is useful for disease diagnosis, risk stratification, treatment planning, and prediction oftreatment durability. To quantify the dynamic surface geometry of tubular-shaped anatomic structures, we propose a simple,rigorous Lagrangian cylindrical coordinate system to monitor well-defined surface points. Specifically, the proposed systemenables quantification of surface curvature and cross-sectional eccentricity. Using idealized software phantom examples, wevalidate the method’s ability to accurately quantify longitudinal and circumferential surface curvature, as well as eccentricity andorientation of eccentricity. We then apply the method to several medical imaging data sets of human vascular structures toexemplify the utility of this coordinate system for analyzing morphology and dynamic geometric changes in blood vesselsthroughout the body.

Keywords Vascular system . Cylindrical coordinates . Lagrangian . Surface curvature . Eccentricity

1 Introduction

Accurate description of vascular geometry is important forunderstanding vascular anatomy, physiology, and pathology.Morphologic changes to vascular anatomy are commonly as-sociated with clinical diagnosis and assessment. In the case ofaortic aneurysms, for example, the simple measurement ofaneurysm diameter can be a predictor of rupture, and aneu-rysm volume has been shown to correlate with the risk ofclinical sequelae [3, 14]. Moreover, descriptions of dynamicanatomy, in the form of vascular deformations due to cardiac,respiratory, and musculoskeletal influences, can provide fur-ther insight into the physiological and pathological processesassociated with disease development. For example, radial

aortic compliance can help characterize degenerative diseasein the aorta and lower extremities [4, 5]. Axial deformationand elasticity of the superficial femoral artery can be used asan indicator of lower extremity arterial health and a predictorof stent fracture [6]. Also, degree of in vivo deformation ofimplanted stents provides insight to predict long-term perfor-mance of the stents inside stenotic arteries [20].

Three-dimensional (3D)medical imaging is capable of pro-viding exquisite geometric information, from which 3D geo-metric models can be constructed. These geometric modelscan then be used to quantify vascular deformation for deviceevaluation and development as well as perform hemodynamicand vessel structure simulations [2, 6, 8–10, 12, 13, 21–24].One of the most established 3D lumen modeling methods isbased on centerline construction, orthogonal 2D segmenta-tions, and surface lofting [28]. While these methods allowfor analysis of motion and deformation of lumen centerlinesand cross sections, they lack the ability to robustly and fullycharacterize 3D vascular surface geometry. For example, theycannot fully quantify the variation in surface curvature along ahighly curved vessel, such as in the aortic arch.

Developing more nuanced methods to quantify 3D geo-metric and morphological features of the human vascular sys-tem, and their dynamic changes, is needed to better understandhow devices interact with the vascular system and the biome-chanical characteristics that determine a patient’s prognosis

* Torbjörn Lundhtorbjorn.lundh@chalmers.se

1 Division of Vascular Surgery, Stanford University, Stanford, CA,USA

2 Department of Mathematical Sciences, Chalmers University ofTechnology and University of Gothenburg, 41296 Gothenburg, Sweden

3 Department of Bioengineering, Stanford University, Stanford, CA,USA

Medical & Biological Engineering & Computing (2018) 56:1659–1668https://doi.org/10.1007/s11517-018-1801-8

and potential response to treatment. For example, recent ef-forts in surface modeling and analysis have demonstrated ex-cellent promise for better predicting aortic aneurysm rupture[12, 16, 18, 19]. However, because the key parameter forevaluating mechanical fatigue of a medical device is basedon alternating strain at a particular material points, aLagrangian-basedmethod to quantify deformation is also war-ranted. Here we present a method for creating a Lagrangiandescription of approximate cylindrical structures based on acylindrical coordinate system. Building from a vessel center-line and lumen cross-sectional contours, this coordinate sys-tem can describe complex surface geometry, including longi-tudinal and circumferential curvature, cross-sectional eccen-tricity, and the orientation of eccentricity. We validate thismethod with idealized software phantoms and demonstratethe wide potential of this method by analyzing dynamicchanges of blood vessel geometry in patient-specific examplesof the thoracic aorta, abdominal aorta, and iliofemoral vein.

2 Methods

2.1 Formulation

To develop a Lagrangian cylindrical coordinate system whichcan accurately quantify the surface of a tubular anatomicalstructure, the first step is to develop a continuous coordinatesystem for the longitudinal and angular dimensions. The lu-minal surface of a tubular structure, such as a blood vessel, canbe defined by a sequence of n cross-sectional contours Si, in3D Cartesian space (Fig. 1a). These contours, formally denot-

ed as Sif gni¼1, are defined as Si ¼ bij� �mi

j¼1 where bij = (xij, yij,

zij) defines the mi individual points on each contour. For eachcontour, the centroid Ci ¼ xi; yi; zið Þ can then be used toconstruct a centerline curve Cif gni¼1. The arc length of thiscenterline curve, σ, is used as the longitudinal coordinate forthis coordinate system.

Next, the angular coordinate, θ, can be defined by specify-ing a particular circumferential reference point among theboundary points bij for each contour Si, which is called γi(Fig. 1b). To define this reference point γi in a consistent,

non-arbitrary way, an initial material point must be identifiedwith a fiducial marker. For vascular structures, a bifurcationpoint serves as a natural fiducial marker. The lumen bound-aries of the mother and daughter branches are defined as twoseparate sets and the most distal intersection of these two setsis selected as the material point δ and will be the landmark onthe mother vessel that will serve as the natural reference point(Fig. 2). Then, the closest boundary point to this landmark isdenoted the Greenwich point γ of the vessel. That is, γ ∈ {bij}where ∣δ − γ ∣ =mini, j ∣ δ − bij∣. Now, suppose that γ is se-lected on the kth contour Sk. This point is used as the referencepoint for the centerline curve, such that C(0) =Ck and C(σ) isproximal to Sk if σ < 0 and distal to Sk if σ > 0. Correspondingreference points on every contour can then be assigned byletting γk = γ, and defining γk + 1 using a projection onto theSk + 1 section as explained in more detail later. Iterating thisprocess defines Greenwich points for each contour. The piece-wise linear curve defined by these points is defined as theGreenwich curve.

Using the longitudinal σ and angular θ dimensions, theLagrangian cylindrical surface function r(σ, θ) is created(Fig. 1b). By using linear interpolation in both dimensions, acontinuous vessel coordinate system provides a Lagrangiandescription of every vessel boundary point. The radial func-tion r(σ, θ) can be expanded to include a time parameter, i.e.,r(σ, θ, t), in order to define changes of the vessel surface withrespect to time.

This Lagrangian coordinate system can be used to quantifyand monitor the surface curvature in both the circumferential(θ) and longitudinal (σ) directions. To compute the curvatureat a given surface point, a circle is fit around the initial pointand two of its symmetric neighboring points in the direction ofinterest (Fig. 3a, b). The reciprocal of the radius of this circle isdefined as the magnitude of the curvature. The product andmean of the circumferential and longitudinal curvatures at agiven point could be used to serve as proxies for the Gaussiancurvature and mean curvature, respectively.

Additionally, this system can be used to quantify both theorientation and magnitude of the eccentricity of the structure.Classically, the eccentricity of an ellipse with major axis a and

minor axis b, defined by x2a2 þ x2

a2 ¼ 1, has eccentricityffiffiffiffiffiffiffiffiffi1− b2

a2

q.

Fig. 1 Definition of a contours Si,contour centroids Ci, andGreenwich points γi along thelongitudinal axis of the tubularstructure and b the Greenwichpoint γi on each section Si fromwhich the angular dimension θcan be defined

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This definition can be generalized to irregular contours byselecting the two orthogonal diameters through the centerpoint Ciwhile minimizing the quotient dD whereD is the majoraxis and d is the axis perpendicular to D and defining the

eccentricity asffiffiffiffiffiffiffiffiffiffi1− d2

D2

q. The direction of eccentricity is defined

as the positive angle θe between the major diameter D and theGreenwich point γi (Fig. 3c). By tracking the direction ofeccentricity as a function of σ, the static and dynamic spiralityof eccentricity can be quantified.

2.2 Optimization of methods

In order to apply this method to arbitrary complex tubularstructures, derivation of the Greenwich curve and pre-scribed window sizes to compute curvature need to beoptimized. For this optimization exercise, two idealizedsoftware phantoms were utilized. The first was a simpletubular phantom with circular contours and a longitudinalbend causing variation in longitudinal curvature (Fig. 4a).The second phantom was designed with non-circular cross

sections and two bends in different planes, exhibiting var-iable circumferential curvature, longitudinal curvature, ec-centricity, and orientation of eccentricity (Fig. 4b).

For defining the points that comprise the Greenwich curve,three methods were attempted. The first method started withthe initial point γk and projected the vector γk −Ck onto theSk + 1th section along the vector Ck + 1 −Ck. Then, γk + 1 is se-

lected to be the closest point in the sequence b kþ1ð Þ j� �mkþ1

j¼1:

The second method instead projected the vector γk −Ck alongthe normal of the Skth section onto the Sk + 1th section anddefined γk + 1 in the same way. The third method simply se-

lected γk + 1 to be the closest point in b kþ1ð Þ j� �mkþ1

j¼1from γk.

Using any of these methods, Greenwich points on every sec-tion can be defined. The piecewise linear curve formed bythese Greenwich points is then designated as the Greenwichcurve. In this paper, we selected the first method, i.e., projec-tion of the vector γk −Ck onto the Sk + 1th section along thevector Ck + 1 −Ck, since we would like to obtain a Greenwichcurve that more robustly tracks the centerline curve.

To determine the optimal span length of the three points tocalculate curvature (i.e., window size), the idealized

Fig. 2 Location of the Greenwichpoint δ on a CT-based anatomicmodel. Orthogonal cross-sectionsto the mother vessel centerline acompletely proximal to thedaughter vessel where thedaughter vessel is not visible, b atthe daughter vessel where thedaughter vessel appears as aBbud,^ c at the most distalintersection of the mother anddaughter vessels where theintersection point is chosen as δ,and d completely distal to thedaughter vessel where the motherand daughter vessels are separate

Fig. 3 Definition of alongitudinal curvature (outercurve in blue and inner curve inred), b circumferential curvature,

and c eccentricity,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−d2=D2

q,

and orientation of eccentricity (θe)with respect to the Greenwichpoint γi, where d/D is minimizedto find the major and minor axes(color figure online)

Med Biol Eng Comput (2018) 56:1659–1668 1661

phantoms, where the analytic curvatures were known, wereused. A variety of window sizes were evaluated for both thelongitudinal and circumferential directions and window sizeswhich adequately resolved the true curvature, yet did not pro-duce substantial spurious oscillations, were selected.

2.3 Application to phantoms and human data sets

The optimized coordinate systemwas applied to the two phan-toms described above, as well as three human data sets toexemplify the full range of quantifications and analyses pos-sible. For the human data, high-resolution computed tomog-raphy (CT) imaging data were acquired of a thoracic aorticendograft implanted via thoracic endovascular aortic repair(TEVAR) (Fig. 5a), an abdominal aorta and the visceral arterybranches after endovascular aneurysm repair (EVAR) (Fig.5b), and the iliofemoral veins after stent implantation (Fig.5c). From the idealized phantoms and the human data sets,

centerlines and orthogonal segmentations were created usingSimVascular (Open Source Medical Software Corporation,San Diego, CA) [28]. Modeling was started by creating aninitial lumen path manually along the center of vessel lumens.Then, semi-automatic 2D level set segmentation was per-formed orthogonally to this initial path, at every one half ra-dius of the vessel. From every segmentation contour, mathe-matical centroid was extracted and connected to form the cen-terline. To assure the orthogonality of lumen contours, subse-quent round of 2D segmentation was performed along thecenterline instead of initial path, with consistent interval.The centerline was updated according to the second-roundcontours, which together used as the input for this application.The cylindrical coordinate system methods were applied toquantify the longitudinal and circumferential curvature, aswell as the amplitude and direction of eccentricity at eachcross section. Finally, for the thoracic aortic endograft, dy-namic changes in geometry due to cardiac pulsation are shown

Fig. 4 3D models (top) andschematics (bottom) of idealizedsoftware phantoms of a a simplephantom with a 90° bend anduniform circular cross sectionsand b a complex phantom withtwo 90° bends in different planesand varying non-circular crosssections

Fig. 5 Human data sets: 3D high-resolution computed tomography data of a a thoracic aorta, b abdominal aorta, and c iliofemoral veins after endograftimplantation

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by comparing the surface geometries between systole anddiastole.

3 Results

3.1 Optimization of parameters

Figure 6 illustrates a subset of the trials run to optimize win-dow sizes to calculate the surface curvature in the longitudinal(Fig. 6a–c) and circumferential (Fig. 6d–f) directions. For lon-gitudinal curvature, a window size of 20 mm produced spuri-ous noise, as indicated by the deep ridges in Fig. 6a, while awindow size of 40 mm underestimated the peak longitudinalcurvature of 1 cm−1 on the inner curve of the second bend by7% (Fig. 6c). A longitudinal window size of 30 mm, however,did not exhibit spurious noise and was able to calculate thepeak curvature to within 0.5% (Fig. 6b). In the circumferentialdirection, a window size of π/8 exhibited spurious oscillations(Fig. 6d) while a window size of 3π/8 underestimated the peakcircumferential curvature of 2 cm−1 on the elliptical crosssections by 35% (Fig. 6f). In this case, a window size ofπ/4 rad (Fig. 6e) was determined to be optimal. Similar rela-tive window sizes were utilized for analysis of idealized phan-tom and human data sets.

3.2 Application to idealized software phantoms

The constructed 3D model of the simple phantom with 2Dcross-sectional contours, centerlines, and Greenwich curve is

shown in Fig. 7a. The pointwise measurements of the longi-tudinal and circumferential curvature along the entire surfaceare shown in Fig. 7b, c. As shown in Fig. 7b, the longitudinalcurvature was calculated to be 0 cm−1 proximal and distal tothe bend and a maximum value of approximately 0.5 and0.25 cm−1 along the inner and outer curve of the bend, respec-tively. In Fig. 7c, the circumferential curvature was calculatedto be approximately 1.0 cm−1 along the entire surface.

Likewise, the constructed model of the complex phan-tom is shown in Fig. 8a and the results of the analyses areshown in the remainder of Fig. 8. Surface calculations oflongitudinal and circumferential curvature are shown inFig. 8b, c, and the quantification of magnitude and orien-tation of eccentricity are shown in Fig. 8d, e. As shown inFig. 8b, the straight sections exhibit 0.0 cm−1 longitudinalc u r v a t u r e , t h e o u t e r c u r v e o f t h e f i r s t b e n dexhibits 0.5 cm−1 curvature, and the inner curve of thesecond bend exhibits 1.0 cm−1 curvature. Figure 8c depictscircumferential curvatures of 1.5 cm−1 along the vertices ofthe major of the elliptical cross sections, approximately0.25 cm−1 along the flat sections of the elliptical crosssections and 1.0 cm−1 at the circular cross sections. InFig. 8d, the magnitude of eccentricity along the longitudi-nal direction was calculated as 0.87 for the proximal anddistal straight sections, with a steep dip at σ = 96 mm,reflecting the location of the circular cross sections. InFig. 8e, the orientation of eccentricity was calculated as1.6 and 0.0 rad for the proximal and distal straight sections,respectively, with a sudden transition at σ = 96 mm. Ascomplement to these images, Table 1 shows the absolute

Fig. 6 Illustration of window size optimization for calculating surfacecurvature of the complex idealized software phantom shown in Fig. 4b:the pointwise calculations of longitudinal curvature for three differentwindow sizes in the sigma direction, a 20 mm, b 30 mm, and c 40 mm,

and the pointwise calculations of circumferential curvature for threedifferent window sizes in the theta direction, d π/8 rad, e π/4 rad, and f3π/8 rad

Med Biol Eng Comput (2018) 56:1659–1668 1663

difference between the numerically estimated curvaturesand the analytic solution with respect to minimum, maxi-mum and median.

3.3 Application to human image data sets

Figure 9a–c illustrates orthogonal segmentations, centerlines,and derived Greenwich curves for each human image data set.Utilizing the coordinate systemmethod on these examples, theamplitude and orientation of the vessel eccentricity, andpointwise longitudinal and circumferential surface curvatureswere quantified. Examples of these analyses can be seen in

Fig. 9d–f. Figure 9d shows the amplitude and orientation ofthe eccentricity of the thoracic aortic endograft (from Fig. 5a).Figure 9e shows the pointwise circumferential surface curva-ture of the aneurysmal abdominal aorta (from Fig. 5b).Figure 9f shows the pointwise longitudinal curvature of theiliac vein (from Fig. 5c).

To demonstrate the ability of the developed method tocompare dynamic changes of surface geometry, Fig. 10depicts a coordinate system analysis of pointwise longi-tudinal surface curvature for the thoracic aortic endograft(from Fig. 5a) at systolic and diastole phases of thecardiac cycle.

Fig. 7 Results of the analyses for the simple idealized phantom from Fig.4a, including a 2D cross-sectional contours and centerlines of the mainvessel and branch (blue) and Greenwich curve (thin black line), blongitudinal curvature mapped pointwise onto the surface of the 3D

structure, and c circumferential curvature mapped pointwise onto thesurface of the 3D structure (the window sizes for the curvaturecomputations are 30 mm for longitudinal and π /4 rad forcircumferential) (color figure online)

Fig. 8 Results of the analyses for the complex idealized phantom fromFig. 4b, including a 2D cross-sectional contours (blue) and centerlines(yellow) of the main vessel and branch and Greenwich curve (thin blackline), b longitudinal curvature mapped pointwise onto the surface of the3D structure, c circumferential curvature mapped pointwise onto the

surface of the 3D structure, d magnitude of eccentricity along thecenterline arc length, and e orientation of eccentricity along thecenterline arc length (the window sizes for the curvature computationsare 30 mm for longitudinal and π/4 rad for circumferential) (color figureonline)

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4 Discussion

In this paper, we describe a robust method for fully describingthe surface geometry of complex anatomic tubular structuresby using a Lagrangian cylindrical coordinate system. Themethod was validated on idealized software phantoms andthen applied to three human data sets of different anatomies.

To ensure a robust and widely applicable system, windowsizes for curvature calculation need to be standardized basedon the anatomy of interest. The window size needs to be smallenough to ensure accurate estimation of maximum curvaturevalues, yet not so small as to cause substantial spurious oscil-lations. Based on experiments on the complex idealized phan-tom in Fig. 4b, the optimal window size for longitudinal andcircumferential curvature calculations was 30 mm andπ/4 rad, respectively. These correspond roughly the diameter

of the vessel and one eighth of the circumference, respectively.These guidelines were used for all subsequent analyses.

Application to the simple and complex idealized phantomsfrom Fig. 4 shows excellent agreement with analytic solutionsof surface curvature and eccentricity. Figure 7 shows that thestraight sections of the simple phantom were correctly calcu-lated to have 0 cm−1 longitudinal curvature and that the innerand outer curvatures at the bend correctly corresponded with 2and 4 cm radius of curvature, respectively. In addition, thecircumferential curvature was correctly calculated to corre-spond with 1 cm radius of curvature (2 cm diameter).

Analysis of the complex idealized phantom had similarexcellent correspondence with analytic solutions, such as lon-gitudinal curvature values of 0.5 and 1.0 cm−1 at the outercurve of the first bend and inner curve of the second bend,respectively (Fig. 8). For the contour cross sections, the

Table 1 Absolute curvaturedifference between numericalestimation and analytic solution

Absolute curvature difference (cm−1) Minimum Maximum Median

Simple phantom, longitudinal curvature 5.7 × 10−8 0.53 8.8 × 10−4

Simple phantom, circumferential curvature 1.9 × 10−4 0.24 0.0094

Complex phantom, longitudinal curvature 1.9 × 10−6 1.0 0.13

Complex phantom, circumferential curvature 0.00091 1.7 0.47

DiastoleSystole

Eccentricity

Orienta�on of eccentricity

(mm)

(rad)

(mm)

Circumferen�al curvature at inspira�on Longitudinal curvature at supine

b c

d e f

(cm

-1)

(cm

-1)

a

Fig. 9 Results of the analyses for the human CT image data from Fig. 5,including 3D representations of the a thoracic aortic endograft duringdiastole, b aneurysmal abdominal aorta during inspiration, and c rightiliofemoral vein in the supine position, the corresponding d magnitude

(above) and orientation (below) of the eccentricity along the centerline ofthe thoracic aortic endograft, e circumferential curvature of the abdominalaorta, and f longitudinal curvature of the right iliofemoral vein

Med Biol Eng Comput (2018) 56:1659–1668 1665

circumferential curvatures calculated for the circular cross sec-tions and the flat sections of the elliptical cross sections were1.0 and 0.25 cm−1, respectively, which perfectly match theanalytic solutions. The calculated circumferential curvatureof 1.5 cm−1 at the vertices of the ellipse is 25% below theanalytic solution; however, increased point sampling aroundthe contour would greatly improve this calculation. The ana-lytic eccentricity of an ellipse with a 4 cm major diameter and

2 cm minor diameter isffiffiffi3

p=2 (dotted pink line in Fig. 8d),

which is almost exactly the value calculated by our method. Inaddition, the orientation of eccentricity calculated by themethod closely approximated the π/2 and 0 rad analytic solu-tions (Fig. 8e). Both eccentricity graphs capture the transitionpoint at the correct longitudinal location where the idealizedmodel bent from one plane to an orthogonal plane.

Figure 9 shows applicability of the method to actual med-ical images of different vascular structures. For the thoracicaortic endograft, the eccentricity was calculated to be higher(less circular) at the ends of the graft, which makes sense dueto the decrease in hoop strength at the free ends of the stentgraft (Fig. 9d). In the abdominal aorta example, circumferen-tial curvature calculations show higher curvatures at the nor-mal proximal aorta (equivalent to diameter ≈ 2 cm), lowercurvatures at the aneurysm (equivalent to diameter ≈ 6 cm),and moderate curvature at the transition between normal andaneurysmal sections (Fig. 9e). For the iliofemoral vein, thelongitudinal curvature was predominantly 0.2 cm−1 or equiv-alent to approximately 5 cm radius of curvature (Fig. 9f).

The CT images and 3D geometric models of the thoracicaortic endograft in Fig. 10a–d illustrate how difficult it is toqualitatively identify subtle differences in surface geometryand morphology of vessels at different physiological states.However, with the color map of pointwise longitudinal

surface curvature (Fig. 10e, f), the coordinate system analysisshows the power of its quantitative sensitivity. For example,the diastolic phase (Fig. 10f) shows a larger region of highlongitudinal curvature along the inner curve of the proximaldescending aorta as compared to the systolic phase (Fig. 10e).This makes sense because the during the high pressure pulseof systole, we would expect the endograft to straightenslightly.

Several methods have been developed to model humanvessels and quantify geometric curvature. Classic methodshave employed 2D level set segmentation creating a set oforthogonal contours following vascular lumen, acquiring cen-terlines from the contours, and computing the curvature alongthe centerline [6, 7, 21, 22, 24, 25]. Other available methodsinclude 3D segmentation with growing seeds, acquiring cen-terline from inscribing spheres in lumenal surface, and com-puting the centerline curvature [11, 15]. It is also possible toacquire surface curvature directly across 2D surface patchessuch as a built-in function provided by the Vascular ModelingToolkit (VMTK) [1]. The method presented in this study im-proved the classic centerline-based method by computing thesurface curvature using 2D-segmented contours as an input. Inaddition to the traditional centerline curvature, we believe thatsurface curvature provides additional, relevant information forcharacterizing vascular dynamics and designing novel medi-cal devices. Furthermore, the method in this study providescurvature output along surface lines, not 2D surface patcheslike the VMTK methods above. The main reason we comput-ed surface curvature was to quantify inner and outer line ge-ometries. These surface line curvatures are crucial to under-stand how medical device dynamically deform under in vivocondition and if in-stent restenosis occurs along the surfaceline with challenging geometry [29]. The methods computing

e

f

(cm-1)g

a b

c d

0.6

(cm-1)

0.5

0.4

0.30.3

0.2

0.1

0

0.2

0.1

Fig. 10 Illustration of thoracic aortic endograft in systole (top row) anddiastole (bottom row) (the human CT image data from Fig. 5a). Leftcolumn shows CT images for a systole and b diastole; middle columnshows 3D models of the cylindrical coordinate system for c systole and ddiastole. The next column shows pointwise longitudinal curvature of theendograft surface for e systole and f diastole. Right column shows the

pointwise absolute difference in longitudinal curvature between systoleand diastole (g). Note how while it is difficult to visualize differences inlongitudinal curvature by looking at the CT images or 3D models, thecolor maps of curvature show quantitative differences. According to thepresented example of longitudinal curvature changes, cardiac pulsationinduces curvature changes along the most distal aorta

1666 Med Biol Eng Comput (2018) 56:1659–1668

surface curvature across 2D surface patches can also calculatesurface line curvature with secondary calculations and aver-aging, but this requires additional steps and relies on averag-ing across 2D areas which results in lower resolutioncalculations.

More complete characterization of vascular geometry mayhelp predict disease severity, such as aneurysm rupture risk[12, 16, 18, 19, 26], quantify pre- and post-operative geomet-ric alterations to determine the mechanical impact of deviceson the native anatomy [25], establish boundary conditionswith which to evaluate and predict device failures due to cy-clic fatigue [7], and more fully describe dynamic anatomy tocome up with better device solutions. For example, in vivoarterial motions have often been implicated in cyclic mechan-ical fatigue and stent fracture [6, 7, 13, 17, 21]. Additionalaortic endograft design challenges for aneurysm or dissectionrepair include cardiac-induced deformation, hemodynamicforces, and vulnerable and complex anatomies [27]. TheLagrangian coordinate system described in this paper im-proves our ability to evaluate deformations of anatomy andimplanted devices at material points.

Because this cylindrical coordinate system method usespiecewise linear centerlines with linear interpolation betweencross-sectional contours, the quality of the geometric modeland analyses is dependent of the quality and quantity of thecross-sectional contours. Specifically, the coordinate systemrequires that the original 2D contours to be sufficiently orthog-onal to the centerline and sufficiently densely-spaced.However, this method can include spline interpolation be-tween centerline points and cross-sectional contours, relievingsome of the need for densely packed contours. In addition, themethod can be generalized to input a volumetric model, de-rived from any number of segmentationmethods, and then usethat model to create arbitrary cross-sectional contour densitiesbased on need. These method extensions, along with applyingthese techniques to a wide range of anatomic structures in-cluding vascular, pulmonary, gastrointestinal, and reproduc-tive, will be topics of future research.

Acknowledgements The first author is grateful for sabbatical fundingfrom Barbro Osher Endowment. The authors acknowledge John Kim,David Zhu, and Kelsey Hirotsu for their support on SimVascularmodeling.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict ofinterest.

Open Access This article is distributed under the terms of the CreativeCommons At t r ibut ion 4 .0 In te rna t ional License (h t tp : / /creativecommons.org/licenses/by/4.0/), which permits unrestricted use,distribution, and reproduction in any medium, provided you give appro-priate credit to the original author(s) and the source, provide a link to theCreative Commons license, and indicate if changes were made.

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Torbjörn Lundh is a professor inma thema t i c s a t Cha lme r sUniversity of Technology inSweden in the field of biomathe-matics and the vice president ofthe European Soc i e ty fo rMathematical and TheoreticalBiology

Ga-Young is a consulting profes-sor of surgery at StanfordUniversity. Her research includesanalyzing complex geometry ofhuman vasculature in 3D and cor-relating to clinical outcome andlong-term performance of medi-cal devices.

Phillip DiGiacomo is a graduatestudent in Bioengineering atStanford University. He is inter-ested in the use of multimodalmedical imaging technologiesand computational tools to probethe physiological basis of diseasestates.

Christopher Cheng is a profes-sor of surgery at StanfordUniversity and has over 20 yearsof experience in academic re-search and the medical device in-dustry. His research has coveredhemodynamics and vascularstructure mechanics.

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